Immediate versus Eventual Conversion: Comparing Geodetic and Hull Numbers in P 3-Convexity

We study the graphs G for which the hull number h(G) and the geodetic number g(G) with respect to P3-convexity coincide. These two parameters correspond to the minimum cardinality of a set U of vertices of G such that the simple expansion process that iteratively adds to U , all vertices outside of U that have two neighbors in U , produces the whole vertex set of G either eventually or after one iteration, respectively. We establish numerous structural properties of the graphs G with h(G) = g(G), which allow the constructive characterization as well as the efficient recognition of all triangle-free such graphs. Furthermore, we characterize the graphs G that satisfy h(H) = g(H) for every induced subgraph H of G in terms of forbidden induced subgraphs.